Conjugation (group theory): Difference between revisions
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imported>Richard Pinch (new entry, just a placeholder really) |
imported>Richard Pinch (relate to commutator, conjugacy and conjugacy class) |
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:<math>x^y = y^{-1} x y . \,</math> | :<math>x^y = y^{-1} x y . \,</math> | ||
If ''x'' and ''y'' [[commutativity|commute]] then the conjugate of ''x'' by ''y'' is just ''x'' again. The [[commutator]] of ''x'' and ''y'' can be written as | |||
:<math>[x,y] = x^{-1} x^y , \, </math> | |||
and so measures the failure of ''x'' and ''y'' to commute. | |||
Two elements are said to be conjugate if one is obtained as a conjugate of the other: the resulting [[relation (mathematics)|relation]] of ''[[conjugacy]]'' is an [[equivalence relation]], whose [[equivalence class]]es are the ''[[conjugacy classess]]''. | |||
Revision as of 12:58, 15 November 2008
In group theory, conjugation is an operation between group elements. The conjugate of x by y is:
If x and y commute then the conjugate of x by y is just x again. The commutator of x and y can be written as
![{\displaystyle [x,y]=x^{-1}x^{y},\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c04fa1883bb53d3a0894d9dcbd4978ad24611a0)
and so measures the failure of x and y to commute.
Two elements are said to be conjugate if one is obtained as a conjugate of the other: the resulting relation of conjugacy is an equivalence relation, whose equivalence classes are the conjugacy classess.